Finite Group Databasebeta

← #37535365

Subgroups — #37535365

Order 23168.

280 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26041838C211yesmaximal, characteristic
2C2×#19153733C211yesmaximal, characteristic, complemented
3#26041068C212yesmaximal, complemented
4#26041065C212yesmaximal, complemented
5C2×#99×C181C211yesmaximal, characteristic, complemented
6C22×C22⋊D181C411yescharacteristic, complemented
7C2×#15668433—21no—
8#19153799—21no—
9#19153797C411yescharacteristic, complemented
10#19153797C2211yescharacteristic
11#19153788—21no—
12#19153788—21no—
13#19153785—21no—
14#19153784C411yescharacteristic, complemented
15#19153784C2211yescharacteristic
16C2×C22⋊Dic181C411yescharacteristic, complemented
17#19153733—21no—
18#19153733C2212yes—
19C22×D4×C181C2211yescharacteristic
20C2×C22⋊C4×C181—21no—
21#99×C181—21no—
22#99×C181C2212yes—
23C23×D181C2×C411yescharacteristic, complemented
24C2×C22⋊D181—21no—
25C2×C22⋊D181—21no—
26C2×C22⋊D181—21no—
27C2×C22⋊D181—21no—
28C2×C22⋊D181—41no—
29C2×C22⋊D181—41no—
30C22×Dic181—21no—
31C22×Dic181C2×C411yescharacteristic, complemented
32#15668433—22no—
33#15668433—41no—
34C2×#13089678—41no—
35#15668428—21no—
36#15668428—21no—
37C22⋊Dic181—21no—
38C22⋊Dic181—21no—
39C22⋊Dic181—21no—
40C22⋊Dic181C2×C411yescharacteristic, complemented
41C22⋊Dic181C2×C411yescharacteristic, complemented
42#15668426—21no—
43#15668426—21no—
44C24×C181C2×C411yescharacteristic
45C24×C181D411yescharacteristic
46C2×D4×C181—41no—
47C2×D4×C181—41no—
48C2×D4×C181C2×C411yescharacteristic
49C2×D4×C181D411yescharacteristic
50C2×D4×C181D411yescharacteristic
51C2×D4×C181C2311yescharacteristic
52C22×C4×C181—41no—
53C22×C4×C181D411yescharacteristic
54C22⋊C4×C181—22no—
55C22⋊C4×C181—41no—
56C22×D181—21no—
57C22×D181—41no—
58C22×D181—81no—
59C22⋊D181—42no—
60C22⋊D181—41no—
61C22⋊D181—41no—
62C22⋊D181—81no—
63C22⋊D181—81no—
64C2×Dic181—21no—
65C2×Dic181—21no—
66C2×Dic181—21no—
67C2×Dic181—41no—
68C2×Dic181—81no—
69#13089678—42no—
70C23×C181—21no—
71C23×C181—21no—
72C23×C181—21no—
73C23×C181—21no—
74C23×C181—81no—
75C23×C181—81no—
76C23×C181C22⋊C411yescharacteristic
77C23×C181C22⋊C411yescharacteristic
78C23×C181C22⋊C411yescharacteristic
79C23×C181C22×C411yescharacteristic
80C23×C181C2×D411yescharacteristic
81D4×C181—41no—
82D4×C181—41no—
83D4×C181—41no—
84D4×C181—41no—
85C2×C4×C181—21no—
86C2×C4×C181—21no—
87C2×C4×C181—42no—
88C2×C4×C181—82no—
89C2×C4×C181C22⋊C411yescharacteristic
90C2×C4×C181C2×D411yescharacteristic
91C22×C181—21no—
92C22×C181—21no—
93C22×C181—21no—
94C22×C181—21no—
95C22×C181—41no—
96C22×C181—41no—
97C22×C181—41no—
98C22×C181—41no—
99C22×C181—41no—
100C22×C181—42no—
101C22×C181—81no—
102C22×C181—81no—
103C22×C181—81no—
104C22×C181#9911yescharacteristic
105C22×C181#9911yescharacteristic
106C22×C181C2×C22⋊C411yescharacteristic
107C2×D181—41no—
108C2×D181—41no—
109C2×D181—41no—
110C2×D181—82no—
111C4×C181—21no—
112C4×C181—21no—
113C4×C181—82no—
114Dic181—41no—
115Dic181—41no—
116Dic181—41no—
117C2×C181—21no—
118C2×C181—21no—
119C2×C181—21no—
120C2×C181—41no—
121C2×C181—41no—
122C2×C181—41no—
123C2×C181—41no—
124C2×C181C2×#9911yescharacteristic
125D181—81no—
126C181#212111yescharacteristic, complemented, Sylow
127#2121—1811nomaximal, Sylow
128#534—1811no—
129C2×#100—1811no—
130C2×#99D18111yescharacteristic, complemented
131#352—1812no—
132AΣL(1,𝔽16)—1812no—
133C22×D4—1811no—
134C22×D4C2×D18111yescharacteristic
135C2×M16—3621no—
136#123—3621no—
137#123—1811no—
138#123—1811no—
139#121—3621no—
140#121—3621no—
141#120—3621no—
142#120—1811no—
143#120—1811no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—1811no—
146#100—3621no—
147#100—1812no—
148#99—21no—
149#99C2×D18112yes—
150C24—1811no—
151C24C4×D18111yescharacteristic
152C24C22⋊D18111yescharacteristic
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—3621no—
156C2×D4—3621no—
157C2×D4—3621no—
158C2×D4—3621no—
159C2×D4—7241no—
160C2×D4—7241no—
161C2×D4C4×D18111yescharacteristic
162C2×D4C22⋊D18111yescharacteristic
163C2×D4D72411yescharacteristic
164C2×D4C22×D18111yescharacteristic
165C22×C4—41no—
166C22×C4—3621no—
167C22×C4—1811no—
168C22×C4D72411yescharacteristic
169M16—3622no—
170M16—7241no—
171C2×C8—7241no—
172C4⋊C4—3621no—
173C4⋊C4—3621no—
174C22⋊C4—22no—
175C22⋊C4—41no—
176C22⋊C4—3621no—
177C22⋊C4—3621no—
178C22⋊C4—3621no—
179C22⋊C4—3621no—
180C22⋊C4—3621no—
181C22⋊C4—1811no—
182C22⋊C4—1811no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—81no—
188C23—81no—
189C23—3621no—
190C23—14481no—
191C23—7241no—
192C23#1566842611yescharacteristic
193C23#1566842611yescharacteristic
194C23#1566842611yescharacteristic
195C23C2×C4×D18111yescharacteristic
196C23C2×C22⋊D18111yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—14481no—
202D4—14481no—
203D4—7242no—
204D4—7241no—
205D4—7241no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—82no—
210C2×C4—3621no—
211C2×C4—3621no—
212C2×C4—3621no—
213C2×C4—14481no—
214C2×C4—7241no—
215C2×C4#1566842611yescharacteristic
216C2×C4C2×D72411yescharacteristic
217C8—7242no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—14482no—
232C22—7241no—
233C22—7241no—
234C22—7241no—
235C22#1915373011yescharacteristic
236C22#1915373011yescharacteristic
237C22C2×#1566842611yescharacteristic
238C4—21no—
239C4—21no—
240C4—82no—
241C4—7241no—
242C4—7241no—
243C4—7241no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—14481no—
252C2C2×#1915373011yescharacteristic
subgroupG/H
centerC2C2×#19153730
commutatorC23×C181C22×C4
FrattiniC2×D4C22×D181
FittingC2×#99×C181C2
socleC2×C181C2×#99
prime radical#26041838C2
cosocleC2×D4×C181C23
hypercenterC2×#99D181
hypocenterC181#2121
LCSG ▷ C23×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#99

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#19153733C2semidirect
3. #26041068C2semidirect
4. #26041065C2semidirect
5. C2×#99×C181C2semidirect
6. C22×C22⋊D181C4semidirect
9. #19153797C4semidirect
14. #19153784C4semidirect
16. C2×C22⋊Dic181C4semidirect
23. C23×D181C2×C4semidirect
31. C22×Dic181C2×C4semidirect
40. C22⋊Dic181C2×C4semidirect
41. C22⋊Dic181C2×C4semidirect
126. C181#2121semidirect
130. C2×#99D181semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260418382
C2×#191537332
#260410682
#260410652
C2×#99×C1812
#2121181